Lecture 12: Introduction to Discrete Fourier Transform Sections 2.2.3, 2.3
Lecture 12: Introduction to Discrete Fourier Transform Sections 2.2.3, 2.3.
Dec 22, 2015
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Lecture 12:Introduction to Discrete Fourier Transform
Sections 2.2.3, 2.3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-8
-6
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-2
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5*sin (24t)
Amplitude = 5
Frequency = 4 Hz
seconds
Review: A sine wave
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-2
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5*sin(24t)
Amplitude = 5
Frequency = 4 Hz
Sampling rate = 256 samples/second
seconds
Sampling duration =1 second
Review: A sine wave signal
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-1.5
-1
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1
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2sin(28t), SR = 8.5 Hz
Review: An undersampled signal
Review: The Nyquist Frequency
•The Nyquist frequency is equal to one-half of the sampling frequency.
•The Nyquist frequency is the highest frequency that can be measured in a signal.
The Fourier Transform
•A transform takes one function (or signal) and turns it into another function (or signal)
•Continuous Fourier Transform:close your eyes if you
don’t like integrals
•A transform takes one function (or signal) and turns it into another function (or signal)
•The Discrete Fourier Transform:
The Fourier Transform
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0
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1 N
n
Niknnk
N
k
Niknkn
eHN
h
ehH
Fast Fourier Transform•The Fast Fourier Transform (FFT) is a
very efficient algorithm for performing a discrete Fourier transform
•FFT principle first used by Gauss in 18??•FFT algorithm published by Cooley &
Tukey in 1965•In 1969, the 2048 point analysis of a
seismic trace took 13 ½ hours. Using the FFT, the same task on the same machine took 2.4 seconds!
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Famous Fourier Transforms
Sine wave
Delta function
Famous Fourier Transforms
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0.5
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Gaussian
Gaussian
Famous Fourier Transforms
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Sinc function
Square wave
Famous Fourier Transforms
Sinc function
Square wave
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-100 -50 0 50 1000
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Famous Fourier Transforms
Exponential
Lorentzian
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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
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0.4
0.6
0.8
1
Effect of changing sample rate
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35
f = 8 Hz T2 = 0.5 s
Effect of changing sample rate
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SR = 256 HzSR = 128 Hz
f = 8 HzT2 = 0.5 s
Effect of changing sample rate
•Lowering the sample rate:▫Reduces the Nyquist frequency, which▫Reduces the maximum measurable
frequency▫Does not affect the frequency resolution
Effect of changing sampling duration
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f = 8 Hz T2 = .5 s
Effect of changing sampling duration
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ST = 2.0 sST = 1.0 s
f = 8 HzT2 = .5 s
Effect of changing sampling duration
•Reducing the sampling duration:▫Lowers the frequency resolution▫Does not affect the range of frequencies
you can measure
Effect of changing sampling duration
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2
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f = 8 Hz T2 = 2.0 s
Effect of changing sampling duration
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ST = 2.0 sST = 1.0 s
f = 8 Hz T2 = 0.1 s
Measuring multiple frequencies
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f1 = 80 Hz, T21 = 1 s
f2 = 90 Hz, T22 = .5 s
f3 = 100 Hz, T2
3 = 0.25 s
SR = 256 Hz
Measuring multiple frequencies
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f1 = 80 Hz, T21 = 1 s
f2 = 90 Hz, T22 = .5 s
f3 = 200 Hz, T2
3 = 0.25 s
SR = 256 Hz
FFT in matlab• Assign your time variables
▫ t = [0:255];• Assign your function
▫ y = cos(2*pi*n/10);• Choose the number of points for the FFT (preferably a power of two)
▫ N = 2048;• Use the command ‘fft’ to compute the N-point FFT for your signal
▫ Yf = abs(fft(y,N)); • Use the ‘fftshift’ command to shift the zero-frequency component to
center of spectrum for better visualization of your signals spectrum▫ Yf= fftshift(Yf);
• Assign your frequency variable which is your x-axis for the spectrum▫ f = [-N/2:N/2-1]/N; - this is the normalized frequency symmetrical
about f0 and about the y-axis
• Plot the spectrum▫ plot(f, Yf)
FFT in matlab• Vary the sampling frequency and see what happens• Vary the sample duration and see what happens
Spectrum of a signal
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
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frequency / f s
Approximate Spectrum of a Sinusoid with the FFT
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